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The Index of Isolated Umbilics on Surfaces of Non-Positive Curvature

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The Index of Isolated Umbilics on Surfaces of Non-Positive Curvature The index of isolated umbilics on surfaces of non-positive curvature Autor(en): Fontenele, Francisco / Xavier, Frederico Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 61 (2015) Heft 1-2 PDF erstellt am: 11.10.2021 Persistenter Link: http://doi.org/10.5169/seals-630591 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch LEnseignement Mathematique (2) 61 (2015), 139-149 DOI 10.4171/LEM/61-1/2-6 The index of isolated umbilics on surfaces of non-positive curvature Francisco Fontenele and Frederico Xavier Abstract. It is shown that if a C2 surface M c M3 has negative curvature on the complement of a point q e M, then the Z/2-valued Poincare-Hopf index at q of either distribution of principal directions on M — {q} is non-positive. Conversely, any non-positive half-integer arises in this fashion. The proof of the index estimate is based on geometric- topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincare-Bendixson theory. Mathematics Subject Classification (2010). Primary: 53A05, 53C12; Secondary: 37C10, 37E35. Keywords. Index of an umbilical point, principal curvature lines, Caratheodory conjecture, Loewner conjecture. To Professor Jorge Sotomayor, on the occasion of his 70th birthday 1. Introduction The distributions of principal directions on a surface in I3, defined on the complement of the umbilical set (i.e., the locus where the principal curvatures coincide), have been the object of intense scrutiny since the early days of differential geometry. For both technical and geometric reasons, most of these investigations were conducted under the hypothesis that the surface is real analytic, or at least of class C3, although one needs only C2 regularity in order for the fields of principal directions to be continuous. The aim of this work is to establish an estimate for the local index of these fields, under a natural curvature restriction, but with optimal regularity: 140 F. Fontenele and F. Xavier Theorem 1.1. Let McM3 be a C2 surface, q e M. Assume that the Gaussian curvature of M is negative at every point of M other than q. Then, the Z/2- valued Poincare-Hopf index at q of either distribution of principal directions on M — {q} is non-positive. We observe that the theorem is sharp. Indeed, if the Gaussian curvature remains negative at q, then the distributions of principal directions extend continuously to M, and so the index is zero. On the other hand, given any negative number j e Z/2 one can construct a surface as in the statement of the theorem, even a minimal one (i.e., with vanishing mean curvature), that has an isolated umbilic of index j ([SX3]). When the surface in question is minimal, the conclusion in Theorem 1.1 can be verified using the holomorphic data in the Weierstrass representation of the surface. The main point of the present work is that, surprisingly, Theorem 1.1 applies to surfaces that are merely C2, and not only to those CM surfaces that are minimal. Since complex analysis is no longer available in this more general setting, new tools have to be introduced in order to estimate the index. Loosely speaking, the replacement for complex analysis is, when properly augmented, the classical qualitative theory of planar dynamical systems. Umbilic points are notoriously elusive geometric objects. For instance, on a surface of positive curvature the index of an isolated umbilic need not be positive. In other words, the "dual" statement of Theorem 1.1 does not hold. Indeed, inverting surfaces satisfying the hypotheses of Theorem 1.1 on suitable spheres - a process that does not change the index of the umbilic -, one can produce surfaces of positive Gaussian curvature exhibiting an umbilic whose index is any prescribed negative half-integer. By analogy with the above mentioned sharpness of Theorem 1.1, one might naively expect that every positive half-integer could be realized as the index of an isolated umbilic on a surface of positive curvature. In stark opposition to this expectation, it is actually predicted that on any sufficiently regular surface, without any curvature restrictions whatsoever, the index of an isolated umbilic should be at most one. This is the well-known local Caratheodory conjecture, also known as the Loewner conjecture. We refer the reader to [Bol], [Bro], [FNB]-[GH], [GS]-[Ham2], [Iva]-[Xav] for a sample of the many works, old and new, on this very challenging problem, as well as on various aspects of the global study of principal foliations. It is worth to point out that, if true, the local Caratheodory conjecture would imply, by the Poincare-Hopf theorem, the Caratheodory conjecture (the statement that any C2 immersion of the 2-sphere into M3 has at least two umbilical points; Isolated umbilics on surfaces of non-positive curvature 141 see [GK] for a solution of this conjecture that has been proposed recently). To put our results in perspective, we re-iterate that Theorem 1.1 is sharp and verifies the C2 version of the local Caratheodory conjecture in a geometrically important special case, but with a stronger conclusion. Thus, Theorem 1.1 represents a contribution to the interface between classical differential geometry and classical dynamical systems that stands on its own, since it cannot be subsumed by the resolution of the Caratheodory conjecture. On the other hand, we hasten to add that there is no expectation that the present method can be used to tackle the said conjecture, given our strong reliance on negative curvature. Over the years, the task of estimating the index of isolated umbilics has proven to be an arduous one, often involving lengthy and intricate arguments (e.g., [Iva]). Against this backdrop, it is pleasing that the proof of Theorem 1.1, albeit delicate in its own right, is rather conceptual. The arguments are based on an index theorem for abstract symmetric tensors on Riemannian surfaces, elements of the classical qualitative theory of two-dimensional dynamical systems, and a modicum of topology and classical differential geometry. Although the main result is new, the subject matter lends itself to a more expository style and, accordingly, full details are provided. 2. An index theorem for abstract symmetric tensors A continuous symmetric tensor field A of type (1,1), defined on a Riemannian surface M, is said to have an isolated A-singularity at q e M if there exists a neighborhood V of q such that the eigenvalues of A(p) are unequal for any p e V — {q}. This condition is equivalent to the requirement that the traceless part A{p) — |(trA(p))I of A{p) be non-zero for all p ^ q. If the eigenvalues of A(q) are distinct, then q is automatically an isolated A-singularity (one then thinks of q as being a "removable" A-singularity). Under the above conditions, there are two continuous line fields on M — {q}, not necessarily orientable, which correspond to the diagonalizing directions of A. Given a continuous field of directions £ on V — {q}, denote by A£ the field of directions obtained by applying A{p) to any vector generating the one- dimensional subspace £(/?) of TPM. We write j(A) for the index at q of either of the two fields of diagonalizing directions of A. Similarly, j(rj) stands for the index at q of a line field rj with an isolated singularity at q. For the more general definition of local index of a totally real n-form (the case n 1 corresponds to a direction field), as well as the proof of the two-dimensional Poincare-Hopf theorem in this context, see [FNB]. 142 F. Fontenele and F. Xavier Theorem 2.1. Let A be a continuous symmetric tensor field of type (7,7) on a Riemannian 2-manifold M, and q e M an isolated A -singularity. Then, for every continuous field of directions £ on a punctured neighborhood of q, one has (2.1) 2j(A) j((A - IM)/) £) + _/(£)• Remarks. One should think of £ as being a "test" line field. For instance, if £ is chosen to be one of the continuous fields of eigendirections of A, each term on the right hand side of (2.1) equals J (A). As another illustration to see that 2 is the correct factor in the left hand side, let A be a continuous symmetric tensor field on a compact orientable surface M of non-zero genus, with the property that the set F\ where A is a multiple of the identity is finite. Let £ be a continuous line field on M with a finite set F2 of singularities. Applying (2.1) around each point in Fi U F2, and summing, one sees from the Poincare-Hopf theorem that both sides of (2.1) equal 2/(M).
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